The dress code for competition is clearly printed in the British Dressage Rule Book which should be used as the definitive guide. The following is however a brief guide.
The Rider:
Preliminary - Advanced Medium level:
The rider can wear shirt / tie or stock. This can be worn with either black / blue jacket or tweed / hacking jacket. You can wear either a hunting cap, bowler hat or a crash cap. Uniform is also permitted.
Advanced level:
Uniform or tail coat with top hat. Alternatively black / blue coat with white / cream stock with the same hats.
Hats: These are compulsory, the BSI Standard hats are recommended but
are not compulsory. Crash cap covers should be black, brown or navy.
If the hat comes off in the test it must be replaced immediately - either
being handed to you or the rider dismounts and replaces it. There is no
penalty and the test re starts at the beginning of the movement in which
it can off.
Breeches / jodhpurs, When worn with an official uniform they can be the colour of that uniform. Otherwise they should be white, beige or cream. Boots should be black or brown. No half chaps are allowed. Gloves must be worn.
Spurs can be worn at any level but are compulsory from Advanced level upwards (this used to be from medium level). Whips may be used at any level but are not permitted in any Regional or Championship or at the Selectors request.
Equipment for the horse:
For a complete reference for which equipment is permissible it is advisable to obtain a copy of the up to date rule book from British Dressage.
A guide can be as follows:
Saddles must be English or Continental. Side saddles are permitted but must notify the organise at time of entry.
The bridle for the Preliminary and Novice level must be of a simple snaffle variety. The double bridle is optional for Elementary to Advanced Medium levels but for Advanced the double is compulsory. The drop, flash or cavesson noseband can be worn with the snaffle, only the cavesson is to be worn with the double bridle.
It is best to study the rule book to ascertain whether any individual bit is allowed.
The only other piece of tack that is allowed is the breastplate. Brushing boots or bandages are fine for use in the warm up but must be removed before the test.